Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Calculate \(\int_{a}^{b} f(x) d x\), where a and \(b\) are the left and right end points for which f is defined, by using the Interval Additive Property and the appropriate area formulas from plane geometry. Begin by graphing the given function. $$ f(x)=\left\\{\begin{array}{ll} -\sqrt{4-x^{2}} & \text { if }-2 \leq x \leq 0 \\ -2 x-2 & \text { if } 0

Step-by-Step Solution

Answer

The integral \( \int_{-2}^{2} f(x) \, dx = -2\pi - 4 \).

Step 1: Graph the Function

First, plot the graph of the piecewise function.- The portion defined by \(-\sqrt{4-x^{2}}\) from \(-2 \leq x \leq 0\) describes the lower half of a circle with radius 2 centered at \(x=0\).- The segment described by \(-2x-2\) from \(0 < x \leq 2\) is a straight line with slope -2 and y-intercept -2.

Step 2: Identify Areas Under the Curve

Identify the geometric shapes under the given piecewise function over the specified interval:- The arc \(-\sqrt{4-x^2}\) forms a semicircle with a radius of 2 along \(-2 \leq x \leq 0\). The area of a full circle is \pi (2)^2 = 4\pi\, so the semicircle's area is \2\pi\, but negative since it is below the x-axis, giving us \(-2\pi\).- The line \(-2x-2\) forms a triangular shape from \(x = 0\) to \(x = 2\) with vertices at \( (0,-2)\) and \( (2,-6)\). The height is 4 and the base is 2, so the area is \( \frac{1}{2} \times 2 \times 4 = 4 \).

Step 3: Apply Interval Additive Property

Calculate the definite integral by summing the areas identified:- The integral from \(-2 \leq x \leq 0\) is \(-2\pi\) from the semicircle.- The integral from \(0 < x \leq 2\) is \(-4\) from the triangle.- Therefore, the total integral is \(-2\pi - 4\).

Key Concepts

Interval Additive Property

The Interval Additive Property is a crucial tool when calculating definite integrals, especially for piecewise functions. It states that if you can break an integral \([\int_{a}^{b} f(x) \, dx]\) into parts, you can add each part to find the total area under the curve.
For instance, if a function changes its definition at a certain point, say at \(c\), within the interval \([a, b]\), we apply this property. The integral over \([a, b] \) becomes the sum of two integrals, \(\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx\).
This approach simplifies calculations since each subinterval can often be tackled with basic geometric principles or simpler integration techniques.
In our exercise, we used the Interval Additive Property by dividing the interval \([-2, 2]\) at the point \(x = 0\), making the integral manageable by considering the semicircle and triangle separately.

Piecewise Function

A piecewise function consists of multiple sub-functions, each applying to a specific interval of the domain.
Understanding how to interpret piecewise functions is key when analyzing their graphs or computing related integrals.
These functions may appear complex, but it's really about understanding each section of the function individually. In the given exercise, the piecewise function is:

• \(f(x) = -\sqrt{4-x^2}\) for \( -2 \leq x \leq 0 \), forming a semicircle.
• \(f(x) = -2x - 2\) for \( 0 < x \leq 2 \), forming a line which outlines a triangle.

Understanding both segments helps in performing the geometric analysis, necessary for evaluating the overall definite integral by examining each piece within its respective interval. Each segment is a simple geometric shape, simplifying the definite integration process.

Graphing Functions

Graphing is a vital skill for understanding and solving problems involving functions. By visualizing the function, we unlock a deeper comprehension of its behavior over different intervals.
With piecewise functions, graphing each segment separately assists greatly:

• The semicircle for \(-\sqrt{4-x^2}\), from \(-2\) to \(0\), helps identify the region contributing to the integral.
• The line for \(-2x - 2\), from \(0\) to \(2\), shows where the function creates a triangular area.

When you graph these components, it's essential to precisely plot key points, recognizing the boundaries and shapes they form.
Overall, graphing transforms abstract algebraic expressions into tangible visual entities, providing a pathway to understanding and calculating the total area under the function's curve.

Geometric Interpretation of Integrals

Interpreting integrals geometrically involves understanding the integral as a measure of area.
In this case, the integral of a piecewise function represents the total area under the curve from \(a\) to \(b\).
For the semicircle \( -\sqrt{4-x^2} \), the geometric interpretation reveals part of a circle. Its area is half the full circle's area because it stretches over half of the circle's span (negative because it's below the x-axis).

• Calculate a full circle's area with \(\pi r^2\) and halve it.
• This conversion provides the area, which contributes \(-2\pi\) to the integral.

On the other side of \(x = 0\), the expression \(-2x - 2\) defines a triangular area. The base and height of this right triangle are easily found:

• Area is calculated with \( \frac{1}{2} \times \text{base} \times \text{height} \).
• This calculation gives \(\-4\) to subtract from the total integral.

Thus, understanding these shapes converts the abstract integral into a problem of simple geometry, clarifying how each section adds to the integral's value.